Dreaming of our loved ones is a common occurrence. Sometimes it is them visiting us in our dreams. Sometimes it is our subconscious mind bringing them forward in memory. Both are completely normal, in my opinion. I’m just gonna go through and answer your questions one at a time and hopefully help you! 
In my opinion, you will simply know – any being that’s not the person you see will feel off or strange in some way. I imagine it’s like when you see someone out and about that looks like someone you know. As soon as you interact with them you’ll know if they’re the person you thought they were. Even though the dream world is more fluid and liminal than the physical plane, I believe that if people visit us in our dreams, they will present to us in ways we will be familiar with.
Is there any reason she has to move on besides you wanting her to? Just because she is a spirit now rather than a physical being doesn’t necessarily mean she doesn’t have things she wants, too. I don’t believe we can move spirits on if they don’t want to, because they have to be ready for the next stage of life, whatever that is. However, I do believe that you can (and should!) set boundaries during your meditations. Telling her that you love her and that you want her to move on isn’t enough of a boundary to keep her out during your meditations. You may need to use some sort of protection before your meditations to keep her out. This can be crystals you wear, an incantation, or even casting a circle.
Another thought I had while typing this up is that maybe she doesn’t want to interrupt your meditations, but she wants your attention in another way. If you give her a space in your home, such as an ancestor altar, and let her know that in your house, this is her home, she may stop interrupting your meditations. An ancestor altar is a place in your house where you honor those who have passed on and are important to you. It doesn’t need to be elaborate or large. It can be a simple shelf with some photos, or even an end table or something similar.
We have plenty of information on ancestors and ancestor altars here on Spells8. Here are some links that might help you!
There are a few ideas I have about your dream. First, it could be your subconscious trying to work through your feelings about your grandmother. I can’t interpret that part for you because that depends on what you think, how you feel, and how you view things.
Second, it could be as you’ve said - a representation of your grandmother’s mind while she was alive. This may have been done unintentionally, or it may have been done intentionally by whomever told you that you can’t stay with her.
Lastly, it could be an intentional dream given to you by your grandmother to help you see and feel how she feels. Everything is dark and gross in that part of her mind, but when you try to leave, everything gets clean and modern. This may be a representation of how Alzheimer’s affected part of her brain but that the rest of her was still there, covered in the muck of the hut she had to stay in.
I refer to my previous answers about knowing if what we’re interacting with is true or not. You can always add protection and wards to your meditations and spaces in your home. This can help keep anything out that you don’t want there.
Also, I don’t believe you can move a spirit on that isn’t ready. It definitely sounds like your grandmother isn’t ready at all. You can keep her out of your meditations with protection and boundaries, or even out of your home entirely by telling her she’s not welcome there.
I hope this has all helped! Maybe the next time you interact with your grandmother, ask her what she wants. Her answer may surprise you!